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Math Help - Prove set theorems

  1. #1
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    Prove set theorems

    For any sets A, B, C in a universe U:

    Prove that:

    If A n B = C n B and A n B' = C n B' then A = C

    the "n" symbol means "intersect"
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  2. #2
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    Hello,
    Quote Originally Posted by mbcsantin View Post
    For any sets A, B, C in a universe U:

    Prove that:

    If A n B = C n B and A n B' = C n B' then A = C

    the "n" symbol means "intersect"
    (A \cap B) \cup (A \cap B')=(C \cap B) \cup (C \cap B')
    A \cap (B \cup B')=C \cap (B \cup B'), by associative law (Algebra of sets - Wikipedia, the free encyclopedia)

    But B \cup B'=U by definition of the complement. And every set intersected with the universe will result in the set itself.

    Hence A=C
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

    (A \cap B) \cup (A \cap B')=(C \cap B) \cup (C \cap B')
    A \cap (B \cup B')=C \cap (B \cup B'), by associative law (Algebra of sets - Wikipedia, the free encyclopedia)

    But B \cup B'=U by definition of the complement. And every set intersected with the universe will result in the set itself.

    Hence A=C
    Thanks!
    But just a question..

    So this is what I'm trying to prove:
    If A n B = C n B and A n B' = C n B' then A=C

    And then in your answer, you have
    (AnB)U(AnB')=(CnB)U(CnB')

    My question is, wouldn't it be
    (A n B) U (C n B) = (A n B') U (C n B') instead??
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    This it true for \left( {\forall X,Y} \right)\left[ {X = \left( {X \cap Y} \right) \cup \left( {X \cap Y'} \right)} \right].

    So
    \begin{array}{rcl}<br />
A & = & {\left( {A \cap B} \right) \cup \left( {A \cap B'} \right)} \\<br />
{} & = & {\left( {C \cap B} \right) \cup \left( {C \cap B'} \right)} \\<br />
{} & = & C \\<br />
\end{array}
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  5. #5
    Moo
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    Quote Originally Posted by mbcsantin View Post
    Thanks!
    But just a question..

    So this is what I'm trying to prove:
    If A n B = C n B and A n B' = C n B' then A=C

    And then in your answer, you have
    (AnB)U(AnB')=(CnB)U(CnB')

    My question is, wouldn't it be
    (A n B) U (C n B) = (A n B') U (C n B') instead??
    Hmm you were given x=y and z=t
    so i made x+z=y+t, which gives (AnB)U(AnB')=(CnB)U(CnB')
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